Is reinforcement learning possible for chatbots?
The problem is indeed that the 'rules' of conversations are not as fixed as the rules for games. However, you can make use of  descriptive formalism from Discourse Analysis, called adjacency pairs. These describe regularities between utterances on a local level, for example greeting/reply, which would match your "Hi, I'm Alice" and "Nice to meet you".

You will need to be able to classify utterances by your chat bot according to a set of possible responses, and then you can see if a valid response is produced for any given utterance. If the user asks a question, then a greeting will not be a good answer, but a statement could be, if it was a response to the question. This is leaving aside the content and focuses merely on the formal characteristics of the utterance.

If you want to know more about the topic, have a look at Conversation Analysis, which is the linguistic field dealing with the subject.